A note on a problem of Erdös and Hajnal

In [5], Erdös and Hajnal formulate the following proposition, which we shall refer to as Φ: If ? is an order-type such that $\text{|?| = ω}{}_2\text{but}\text{ω}{}_2\text{, ω}{}_2{}^1\text{? ?}$, there is ψ ? ?, |ψ| = ω₁, such that $\text{ω}{}_1\text{, ω}{}_1{}^1\text{? ψ}$. In [2], we showed that if V = L, then ?Φ. We do not know if the assumption V = L can be weakened to CH, or if, in fact, Φ is consistent with CH. However, in this note we show that, relative to a certain large cardinal assumption, Φ is consistent with 2^ω = ω₂, so that ?Φ is not provable in ZFC alone. Our proof has an interesting model-theoretic consequence, which we mention at the end.